Prashanth L. A

Soumen Pachal, Mizhaan Prajit Maniyar, Prashanth L. A

We consider the problem of risk-sensitive control in a reinforcement learning (RL) framework. In particular, we aim to find a risk-optimal policy by maximizing the distortion riskmetric (DRM) of the discounted reward in a finite horizon Markov decision process (MDP). DRMs are a rich class of risk measures that include several well-known risk measures as special cases. We derive a policy Hessian theorem for the DRM objective using the likelihood ratio method. Using this result, we propose a natural DRM Hessian estimator from sample trajectories of the underlying MDP. Next, we present a cubic-regularized policy Newton algorithm for solving this problem in an on-policy RL setting using estimates of the DRM gradient and Hessian. Our proposed algorithm is shown to converge to an $ε$-second-order stationary point ($ε$-SOSP) of the DRM objective, and this guarantee ensures the escaping of saddle points. The sample complexity of our algorithms to find an $ ε$-SOSP is $\mathcal{O}(ε^{-3.5})$. Our experiments validate the theoretical findings. To the best of our knowledge, our is the first work to present convergence to an $ε$-SOSP of a risk-sensitive objective, while existing works in the literature have either shown convergence to a first-order stationary point of a risk-sensitive objective, or a SOSP of a risk-neutral one.

Nithia Vijayan, Prashanth L. A

We propose policy gradient algorithms for solving a risk-sensitive reinforcement learning (RL) problem in on-policy as well as off-policy settings. We consider episodic Markov decision processes, and model the risk using the broad class of smooth risk measures of the cumulative discounted reward. We propose two template policy gradient algorithms that optimize a smooth risk measure in on-policy and off-policy RL settings, respectively. We derive non-asymptotic bounds that quantify the rate of convergence of our proposed algorithms to a stationary point of the smooth risk measure. As special cases, we establish that our algorithms apply to optimization of mean-variance and distortion risk measures, respectively.

Nithia Vijayan, Prashanth L. A

We propose policy gradient algorithms which learn risk-sensitive policies in a reinforcement learning (RL) framework. Our proposed algorithms maximize the distortion risk measure (DRM) of the cumulative reward in an episodic Markov decision process in on-policy and off-policy RL settings, respectively. We derive a variant of the policy gradient theorem that caters to the DRM objective, and integrate it with a likelihood ratio-based gradient estimation scheme. We derive non-asymptotic bounds that establish the convergence of our proposed algorithms to an approximate stationary point of the DRM objective.

Nithia Vijayan, Prashanth L. A

We propose two policy gradient algorithms for solving the problem of control in an off-policy reinforcement learning (RL) context. Both algorithms incorporate a smoothed functional (SF) based gradient estimation scheme. The first algorithm is a straightforward combination of importance sampling-based off-policy evaluation with SF-based gradient estimation. The second algorithm, inspired by the stochastic variance-reduced gradient (SVRG) algorithm, incorporates variance reduction in the update iteration. For both algorithms, we derive non-asymptotic bounds that establish convergence to an approximate stationary point. From these results, we infer that the first algorithm converges at a rate that is comparable to the well-known REINFORCE algorithm in an off-policy RL context, while the second algorithm exhibits an improved rate of convergence.

Nirav Bhavsar, Prashanth L. A

We introduce biased gradient oracles to capture a setting where the function measurements have an estimation error that can be controlled through a batch size parameter. Our proposed oracles are appealing in several practical contexts, for instance, risk measure estimation from a batch of independent and identically distributed (i.i.d.) samples, or simulation optimization, where the function measurements are `biased' due to computational constraints. In either case, increasing the batch size reduces the estimation error. We highlight the applicability of our biased gradient oracles in a risk-sensitive reinforcement learning setting. In the stochastic non-convex optimization context, we analyze a variant of the randomized stochastic gradient (RSG) algorithm with a biased gradient oracle. We quantify the convergence rate of this algorithm by deriving non-asymptotic bounds on its performance. Next, in the stochastic convex optimization setting, we derive non-asymptotic bounds for the last iterate of a stochastic gradient descent (SGD) algorithm with a biased gradient oracle.

Vinay Praneeth Boda, Prashanth L. A

While the objective in traditional multi-armed bandit problems is to find the arm with the highest mean, in many settings, finding an arm that best captures information about other arms is of interest. This objective, however, requires learning the underlying correlation structure and not just the means of the arms. Sensors placement for industrial surveillance and cellular network monitoring are a few applications, where the underlying correlation structure plays an important role. Motivated by such applications, we formulate the correlated bandit problem, where the objective is to find the arm with the lowest mean-squared error (MSE) in estimating all the arms. To this end, we derive first an MSE estimator, based on sample variances and covariances, and show that our estimator exponentially concentrates around the true MSE. Under a best-arm identification framework, we propose a successive rejects type algorithm and provide bounds on the probability of error in identifying the best arm. Using minmax theory, we also derive fundamental performance limits for the correlated bandit problem.